
# 2.4: Nonlinear Equations in $$\mathbb{R}^n$$

Here we consider the nonlinear differential equation

\label{nonlinear2}
F(x,z,p)=0,

where
$$x=(x_1,\ldots,x_n),\ z=u(x):\ \Omega\subset\mathbb{R}^n\mapsto\mathbb{R}^1,\ p=\nabla u.$$
The following system of $$2n+1$$ ordinary differential equations is called characteristic system.
\begin{eqnarray*}
x'(t)&=&\nabla_pF\\
z'(t)&=&p\cdot\nabla_pF\\
p'(t)&=&-\nabla_xF-F_zp.
\end{eqnarray*}
Let
$$x_0(s)=(x_{01}(s),\ldots,x_{0n}(s)),\ s=(s_1,\ldots,s_{n-1}),$$
be a given regular (n-1)-dimensional $$C^2$$-hypersurface in $$\mathbb{R}^n$$, i. e., we assume
$$\mbox{rank}\frac{\partial x_0(s)}{\partial s}=n-1.$$
Here  $$s\in D$$ is a parameter from an $$(n-1)$$-dimensional parameter domain $$D$$.

For example, $$x=x_0(s)$$ defines in the three dimensional case a regular surface in $$\mathbb{R}^3$$.

Assume
$$z_0(s):\ D\mapsto\mathbb{R}^1,\ p_0(s)=(p_{01}(s),\ldots,p_{0n}(s))$$
are given sufficiently regular functions.

The $$(2n+1)$$-vector
$$(x_0(s),z_0(s),p_0(s))$$
is called initial strip manifold and the condition
$$\frac{\partial z_0}{\partial s_l}=\sum_{i=1}^{n-1}p_{0i}(s)\frac{\partial x_{0i}}{\partial s_l},$$
$$l=1,\ldots,n-1$$, strip condition.

The initial strip manifold is said to be non-characteristic if
$$\det\left(\begin{array}{llcl}F_{p_1}&F_{p_2}&\cdots & F_{p_n}\\ \frac{\partial x_{01}}{\partial s_1}&\frac{\partial x_{02}}{\partial s_1}&\cdots & \frac{\partial x_{0n}}{\partial s_1}\\ ... & ... & ... & ...\\ \frac{\partial x_{01}}{\partial s_{n-1}}&\frac{\partial x_{02}}{\partial s_{n-1}}&\cdots & \frac{\partial x_{0n}}{\partial s_{n-1}}\end{array}\right)\not=0,$$
where the argument of $$F_{p_j}$$ is the initial strip manifold.

Initial value problem of Cauchy. Seek a solution $$z=u(x)$$ of the differential equation (\ref{nonlinear2}) such that the initial manifold is a subset of $$\{(x,u(x),\nabla u(x)):\ x\in \Omega\}$$.

As in the two dimensional case we have under additional regularity assumptions

Theorem 2.3. Suppose the initial strip manifold is not characteristic and satisfies  differential equation (\ref{nonlinear2}), that is,
$$F(x_0(s),z_0(s),p_0(s))=0$$. Then there is a  neighborhood of the initial manifold $$(x_0(s),z_0(s))$$ such that there exists a unique solution of the Cauchy initial value problem.

Sketch of proof. Let
$$x=x(s,t),\ z=z(s,t),\ p=p(s,t)$$
be the solution of the characteristic system and let
$$s=s(x),\ t=t(x)$$
be the inverse of $$x=x(s,t)$$ which exists in a neighborhood of $$t=0$$. Then, it turns out that
$$z=u(x):= z(s_1(x_1,\ldots,x_n),\ldots,s_{n-1}(x_1,\ldots,x_n),t(x_1,\ldots,x_n))$$
is the solution of the problem.

### Contributors

• Integrated by Justin Marshall.